Governing equation

The governing equations of a mathematical model describe how the values of the unknown variables (i.e. the dependent variables) change when one or more of the known (i.e. independent) variables change.

Differential equation

Physics

The classic governing equations^[1]^[2] in physics that are lectured^[3]^[4]^[5]^[6] at universities are listed below.

balance of mass
balance of (linear) momentum
balance of angular momentum
balance of energy
balance of entropy
Maxwell-Faraday equation for induced electric field
Ampére-Maxwell equation for induced magnetic field
Gauss equation for electric flux
Gauss equation for magnetic flux

The basic equations in classical mechanics together with Maxwell-Faraday and Ampére-Maxwell equations are all balance equations, and as such each of them contains a time-derivative term which calculates how much the dependent variable change with time. For an isolated, frictionless / inviscid system the first four equations are the familiar conservation equations in classical mechanics.

Darcy's law of groundwater flow has the form of a volumetric flux caused by a pressure gradient. A flux in classical mechanics is normally not a governing equation, but usually a defining equation for transport properties. Darcy's law was originally established as an empircal law, but is later shown to be derivable from an approximation of Navier-Stokes equation combined with an empirical composite friction force term. This explains the duality of Darcy's law as a governing equation and a defining equation for absolute permeability.

The non-linearity of the material derivative in balance equations in general, and the complexities of Cauchy's momentum equation and Navier-Stokes equation makes the basic equation in classical mechanics prune to establishing simpler approximations.

Some examples using differential equations are

Biology

A famous example of governing equation within biology, using differential equations, is

Lotka-Volterra equations are predator-prey equations

Sequence of states

A governing equation may also be a state equation, an equation describing the state of the system, and thus actually be a constitutive equation that has "stepped up the ranks" because the model in question was not meant to include a time-dependent term in the equation. This is the case for a model of an oil production plant which on the average operates in a steady state mode. Results from one thermodynamic equilibrium calculation are input data to the next equilibrium calculation together with some new state parameters, and so on. In this case the algorithm and sequence of input data form a chain of actions, or calculations, that describes change of states from the first state (based solely on input data) to the last state that finally comes out of the calculation sequence.

References

^ Fletcher, Clive A.J. (1991). Computational Techniques for Fluid Dynamics 2; Chapter 1; Fluid Dynamics: The Governing Equations. Vol. 2. Berlin / Heidelberg, Germany: Springer Berlin Heidelberg. pp. 1–46. ISBN 978-3-642-58239-4.
^ Kline, S.J. (2012). Similitude and Approximation Theory (2012 ed.). Berlin / Heidelberg, Germany: Springer Science & Business Media. ISBN 9783642616389.
^ Nakariakov, Prof. Valery (2015). Lecture PX392 Plasma Electrodynamics (Lecture PX392 2015-2016 ed.). Coventry, England, UK: Department of Physics, University of Warwick.[1]
^ Tryggvason, Viola D. Hank Professor Gretar (2011). Lecture 28 Computational Fluid Dynamics - CFD Course from B. Daly (1969) Numerical methods (Lecture 28 CFD Course 2011 ed.). Notre Dame, Indiana, US: Department of Aerospace and Mechanical Engineering, University of Notre Dame.[2]
^ Münchow, Physical Oceanographer Ph.D. Andreas (2012). Lecture MAST-806 Geophysical Fluid Dynamics (Lecture MAST-806 2012 ed.). Newark, Delaware, US: University of Delaware.[3]
^ Brenner, Glover Prof. Michael P. (2000). The dynamics of thin sheets of fluid Part 1 Water bells by G.I. Taylor (MIT course number 18.325 Spring 2000 ed.). Cambridge, Massachusetts, US: Harvard University.[4]

[Fletcher1991-1] Fletcher, Clive A.J. (1991). Computational Techniques for Fluid Dynamics 2; Chapter 1; Fluid Dynamics: The Governing Equations. Vol. 2. Berlin / Heidelberg, Germany: Springer Berlin Heidelberg. pp. 1–46. ISBN 978-3-642-58239-4.

[Kline2012-2] Kline, S.J. (2012). Similitude and Approximation Theory (2012 ed.). Berlin / Heidelberg, Germany: Springer Science & Business Media. ISBN 9783642616389.

[Nakariakov2015-3] Nakariakov, Prof. Valery (2015). Lecture PX392 Plasma Electrodynamics (Lecture PX392 2015-2016 ed.). Coventry, England, UK: Department of Physics, University of Warwick.[1]

[Tryggvason2011-4] Tryggvason, Viola D. Hank Professor Gretar (2011). Lecture 28 Computational Fluid Dynamics - CFD Course from B. Daly (1969) Numerical methods (Lecture 28 CFD Course 2011 ed.). Notre Dame, Indiana, US: Department of Aerospace and Mechanical Engineering, University of Notre Dame.[2]

[Münchow2012-5] Münchow, Physical Oceanographer Ph.D. Andreas (2012). Lecture MAST-806 Geophysical Fluid Dynamics (Lecture MAST-806 2012 ed.). Newark, Delaware, US: University of Delaware.[3]

[Brenner2000-6] Brenner, Glover Prof. Michael P. (2000). The dynamics of thin sheets of fluid Part 1 Water bells by G.I. Taylor (MIT course number 18.325 Spring 2000 ed.). Cambridge, Massachusetts, US: Harvard University.[4]

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